hence \(p_1,\ldots ,p_n\)  are P-values for the model (REF ). To describe the behavior of \(p_i\)  under the alternative, we use that \(\log (n)/\lambda ^{\prime }_i \rightarrow 0\)  and \(\lambda ^{\prime }_i-\lambda _i \rightarrow \infty \)  to obtain a “moderate deviation” estimate (see [1]} [2]}) to the random variable
\(\Upsilon _{\lambda ^{\prime }_i} \sim \mathrm {Pois}(\lambda ^{\prime }_i)\)  as
\(-2 \log \left(\mathsf {\bar{P}}(\Upsilon _{\lambda ^{\prime }_i};\lambda _i )\right)\overset{D}{=}\left( Z + \mu _n \right)^2(1+o_{p}(1)),\qquad Z\sim \mathcal {N}(0,1).\) 
